Category Archives: Mathematics

Constructing A Modified Control Chart,The X-Bar And R Values

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Problem 1 (25 points)

The x-bar and R values for 20 samples of size five are shown in the following table. Specifications on this product have been established as 0.550 ± 0.02.

Sample Number

Xbar

R

1

0.549

0.0025

2

0.548

0.0021

3

0.548

0.0023

4

0.551

0.0029

5

0.553

0.0018

6

0.552

0.0017

7

0.55

0.002

8

0.551

0.0024

9

0.553

0.0022

10

0.556

0.0028

11

0.547

0.002

12

0.545

0.003

13

0.549

0.0031

14

0.552

0.0022

15

0.55

0.0023

16

0.548

0.0021

17

0.556

0.0019

18

0.546

0.0018

19

0.55

0.0021

20

0.551

0.0022

Construct a modified control chart with three sigma limits, assuming that if the true process fraction non conforming is as large as 1%, the process is unacceptable.

Suppose that if the true process fraction nonconforming is as large as 1%, we would like an acceptance control chart to detect this out of control condition with probability 0.90. Construct this acceptance control chart and compare it to the chart obtained in part

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Due: 12:30 pm on March 14th
Name : _____________________________________________________________________________
This optional assignment could up to 10 points extra credit to your Exam 2 grade. Please show appropriate work and use terminology and notation correctly. Unless specified otherwise, you should give exact answers. Partial credit will be given sparingly, if at all. Each problem is worth 1 point. Remember that these should help you practice some of the concepts you will be tested over, but this assignment should not be interpreted as a sample exam.
1. Find the curvature of the curve 𝑟𝑟⃗(𝑡𝑡)=<4sin(2𝑡𝑡),4cos(2𝑡𝑡),4𝑡𝑡>.
2. At what point on the curve 𝑥𝑥=𝑡𝑡3,𝑦𝑦=5𝑡𝑡,𝑧𝑧=𝑡𝑡4 is the normal plane parallel to the plane 3𝑥𝑥+5𝑦𝑦−4𝑧𝑧=2?
3. Find the position vector of a particle that has the given acceleration and the given initial velocity and position:
𝑎𝑎⃗(𝑡𝑡)=sin(2𝑡𝑡),𝑣𝑣⃗(0)=1,𝑟𝑟⃗(0)=1
.
.
Name : ______________________________________________________________________
.. Page 2
4. (a) Evaluate the limit:
(b) Evaluate the limit: .
Name : ______________________________________________________________________
.. Page 3
5. Find for .𝑧𝑧=𝑦𝑦𝑦 𝑦 𝑦𝑦(6𝑥𝑥).
6. Find the equation of the tangent plane to the given surface at the specified point.
Name : ______________________________________________________________________
.. Page 4
7. Let and suppose that (x, y) changes from (2, –1) to (2.01, –0.98)
(a) Compute Δz. (b) Compute dz.
8. Use the Chain Rule to find .
.
Name : ______________________________________________________________________
.. Page 5
9. Find the directional derivative of the function 𝑓𝑓(𝑥𝑥,𝑦𝑦)=(𝑥𝑥+6)𝑒𝑒𝑦𝑦at the point P(3, 0) in the direction of the unit vector that makes the angle 𝜃𝜃=𝜋𝜋3 with the positive x-axis.
10. Find and classify the relative extrema and saddle points of the function
𝑓𝑓(𝑥𝑥,𝑦𝑦)=𝑥𝑥2+2𝑦𝑦2+𝑥𝑥2𝑦𝑦+11

This coursework tests your basic financial mathematics and statistical modelling skills, using spreadsheet software (Excel – formulae, financial maths, graphical features, Data Analysis and Solver tools) as well as your awareness of the reality of how financial products work. Your answers are to be presented in an essay/report format, for which you will use a word processor. In writing your report, please:

state and explain all assumptions, on which your answers are based;

clearly indicate your answer/recommendations;

no evidence of use of excel will result in a fail mark for this element of the coursework component of your mark;

support any answers with the appropriate calculations to arrive at the answer;

include selected screens of formulae underlying computed values. Failure to demonstrate you have created appropriate formulations on excel will be severely penalised. Despite the fact that you will be submitting the Excel file as well, your report is a stand-alone document, meaning a reader should not be required to look at the Excel file to understand your analysis, findings and recommendations;

please note that adequate usage of the excel calculations in the report is important. This means that the key data/findings needs to be included in the report and appropriate referencing needs to be done, i.e. the relevant cell/table/range in the relevant tab of the excel file mentioned at the point of the report when it should be consulted.

The report will have a maximum of 10 pages (including any Appendixes; penalties will be applied for longer submissions – you are required to develop your judgement on what is and isn’t important). Ten percent of the total mark is allowed for quality of the presentation and these marks are distributed among the questions.

You will need to submit a Word document with the report (see instructions above) and an Excel file with the calculations.

Also, please note that late submissions will be penalised, no matter how small and irrespective of computer or internet crashes or any other last minute unexpected problem, so make sure you plan your submission allowing enough time to overcome any last minute problems. This includes keeping up-to-date back-ups of your work!

Notes:

This coursework is your own (individual) work. Any student found guilty of plagiarism will be penalised.Standard penalties for late submissions are applicable.

Question 1: (10%)

With the reduction in sales driven by the financial crisis which have not fully recovered yet, Office Trade, a wholesaler of office equipment has been conducting a review of its stock ordering system. The management of the company has asked each of the analysts in its finance department, to look at each of its products, and you were asked to look at book shelving units. Having made your research, you have determined that it is uncertain when the recovery will start, so you have established the following demand scenarios for the next 12 months:

Scenario

Crisis Continues

Slow Recovery

Medium Recovery

Fast Recovery

Probability

20%

40%

35%

5%

Demand

25,000

35,000

50,000

75,000

Considering the cost per order is £30 and the average carrying cost per unit is £2:

Determine the Economic Order Quantity given the data above.

Produce sensitivity analysis assuming a change of up to 10% up or down on each of the factors individually and on all factors simultaneously.

Make a final recommendation to the board of the company, as to the number of units it should include in each order.

Question 2: (15%)

A car manufacturer is looking to reorganise and increase the efficiency of its manufacturing operations, and is currently looking at an engine assembly line, which makes three different engines (1.8, 2.0 and 2.5litre). The prices charged for the engines are £2,500, £2,850 and £3,750 respectively, while the inputs required to make each of the engines are listed in the table below:

Product

Basic

Medium

High

Max. Available

Aluminium

10 units

11 units

12 units

52,000

Other Metals

5 units

7 units

6 units

30,000

Other Materials

6 units

4 units

9 units

28,000

Labour

2 hours

2 ¼ hours

2 ½ hours

10,000

The costs for the inputs are £25 per hour for Labour and £100, £120 and £35 per unit for Aluminium, Other Metals and Other Materials, respectively. There is also a maximum daily demand for the engines, which is 4,000 1.8litre, 3,500 2.0litre and 2,000 2.5litre.

Formulate this problem as a linear program and use Excel’s Solver to arrive at a solution. Write a short report describing your procedure, justify your formulation and give a recommendation to the firm on the best daily production mix.

Question 3: (30%)

The majority of banks, when making decisions on mortgage applications, will look at two indicators: salary and borrowing as a percentage of purchase price. On the first indicator, banks are normally willing to lend 2.5 times one’s salary or 3.25 times joint salary in a joint mortgage application, while currently most banks will lend up to 75% of the property price on their best rate with penalties for higher percentages. John and Julia are getting married and decided to buy a flat to move into once they do and are looking to take on a 25-year mortgage. You have been given the following data:

John’s current salary is £39,000 p.a. and Julia’s is £37,500 p.a. plus a bonus likely to be around £5,000 (based on previous 3 years experience);

Both have jobs where they partly telecommute, so on average each works from home 2 days a week;

Their total savings at the moment are £25,000;

John owns a flat which he plans to sell, and has been advised that he should be able to sell it for £150,000. The mortgage outstanding on this flat is £112,000;

The average price of flats in the area they would like to move into is as follows: studios £150,000; 1-bedroom £220,000; 2-bedroom £325,000; 3-bedroom £450,000; 4-bedroom £600,000

Having contacted a financial adviser at the end of January, he has identified the following as the best available mortgage rates:

Repayment fixed rate for 2-years of 1.89%. After that period, the rate reverts to the bank’s standard variable rate, which currently is 3.69%;

repayment fixed rate for 5-years of 2.34%. After that period, the rate reverts to the bank’s standard variable rate, which currently is 3.69%;

interest only mortgage at 5% for the life of the loan. In this instance, you would be required to create an investment fund, which pays an interest rate of 3.9% to cover the repayment of the mortgage.

All the rates above are for loans of up to 75% of the property value. There is an increase of 1.5%age points if borrowing is up to 90% of the property value.

Assess:

What is the maximum John and Julia can borrow while taking advantage of the bank’s best mortgage rate;

The amount you advise them to borrow, given their financial and professional situation;

Which is the best mortgage that John and Julia to take out (assume they take out the amount you recommended in b);

Whether that advice would change if interest rates went up or down by up to three percentage points.

Question 4: (20%)

Garnett plc has seen sales in one of its product lines decline over the last two years. The production is currently subcontracted and any changes require a six month notice, so Garnett has to decide now what to do for their most important advertising period, which is in September every year. The options it has identified are:

Option A – Invest £1million to make small changes to the product design and manufacturing process, which will generate increased cash flows in the short term;

Option B – completely redesign the product and production process, which will have a longer lasting effect on cash flows, but will require an investment of £7million.

Garnett’s required rate of return on investments is 12.5% and the estimated cash flows for the two options are as follows (in ‘000s):

Year

Option A

Option B

1

1,000

300

2

1,500

1,500

3

1,750

3,000

4

500

3,000

5

3,000

6

3,000

7

2,700

8

2,000

9

1,500

10

1,000

Required:

Discuss and compare the different types of investment appraisal methods Garnett can use, including a discussion of the advantages and disadvantages of each.

If Garnett had a rule that all investment projects need to payback within 3 years, what project would be chosen? Comment.

Make a recommendation as to which project should be undertaken.

If Garnett believes there is an opportunity to start exporting its product line to another country once sales are finished in its home country (i.e. from year 5), and it thinks it will be able to generate cash flows of £250,000 in the first year, £750,000 in the second and £1,250,000 in the subsequent four years, would your answer to part c) change? How? (Note: production can’t be further increased in the future if option B is chosen now)

Question 5: (15%)

The table below represents data for the profits, sales, average shop size and number of product lines sold by the 20 branches of a retailing company. You have been asked to analyse the data, using the Data Analysis tool in Excel, and make recommendations, including the following:

Summarise the distribution of profits of the twenty branches and comment on the results, including identification of any particularly good or poorly performing branches.

Identify whether there is evidence that the average number of lines stocked per branch is significantly different from 150.

Identify whether there is a significant difference between the profits of two groups of branches, split by the level of sales, with the threshold being £600,000.

Based on this sample, provide a 98% confidence interval for the profits of the twenty branches and comment on the outcome.

Which three questions you choose to answer are up to you: But you should select questions that are most relevant to your experience and that best reflect your individual circumstances.

Keep in mind

All questions are equal: All questions are given equal consideration in the application review process, which means there is no advantage or disadvantage to choosing certain questions over others.

There is no right or wrong way to answer these questions: It’s about getting to know your personality, background, interests and achievements in your own unique voice.

Questions & guidance

Remember, the personal questions are just that — personal. Which means you should use our guidance for each question just as a suggestion in case you need help. The important thing is expressing who are you, what matters to you and what you want to share with UC.

Required question

Please describe how you have prepared for your intended major, including your readiness to succeed in your upper-division courses once you enroll at the university.

Things to consider: How did your interest in your major develop? Do you have any experience related to your major outside the classroom — such as volunteer work, internships and employment, or participation in student organizations and activities? If you haven’t had experience in the field, consider including experience in the classroom. This may include working with faculty or doing research projects.

If you’re applying to multiple campuses with a different major at each campus, think about approaching the topic from a broader perspective, or find a common thread among the majors you’ve chosen.

Choose to answer any three of the following seven questions:

Describe an example of your leadership experience in which you have positively influenced others, helped resolve disputes, or contributed to group efforts over time.

Things to consider:A leadership role can mean more than just a title. It can mean being a mentor to others, acting as the person in charge of a specific task, or taking lead role in organizing an event or project. Think about your accomplishments and what you learned from the experience. What were your responsibilities?

Did you lead a team? How did your experience change your perspective on leading others? Did you help to resolve an important dispute at your school, church in your community or an organization? And your leadership role doesn’t necessarily have to be limited to school activities. For example, do you help out or take care of your family?

2. Every person has a creative side, and it can be expressed in many ways: problem solving, original and innovative thinking, and artistically, to name a few. Describe how you express your creative side.

Things to consider: What does creativity mean to you? Do you have a creative skill that is important to you? What have you been able to do with that skill? If you used creativity to solve a problem, what was your solution? What are the steps you took to solve the problem?

How does your creativity influence your decisions inside or outside the classroom? Does your creativity relate to your major or a future career?

3. What would you say is your greatest talent or skill? How have you developed and demonstrated that talent over time?

Things to consider: If there’s a talent or skill that you’re proud of, this is the time to share it. You don’t necessarily have to be recognized or have received awards for your talent (although if you did and you want to talk about, feel free to do so). Why is this talent or skill meaningful to you?

Does the talent come naturally or have you worked hard to develop this skill or talent? Does your talent or skill allow you opportunities in or outside the classroom? If so, what are they and how do they fit into your schedule?

4. Describe how you have taken advantage of a significant educational opportunity or worked to overcome an educational barrier you have faced.

Things to consider: An educational opportunity can be anything that has added value to your educational experience and better prepared you for college. For example, participation in an honors or academic enrichment program, or enrollment in an academy that’s geared toward an occupation or a major, or taking advanced courses that interest you — just to name a few.

If you choose to write about educational barriers you’ve faced, how did you overcome or strived to overcome them? What personal characteristics or skills did you call on to overcome this challenge? How did overcoming this barrier help shape who are you today?

5. Describe the most significant challenge you have faced and the steps you have taken to overcome this challenge. How has this challenge affected your academic achievement?

Things to consider: A challenge could be personal, or something you have faced in your community or school. Why was the challenge significant to you? This is a good opportunity to talk about any obstacles you’ve faced and what you’ve learned from the experience. Did you have support from someone else or did you handle it alone?

If you’re currently working your way through a challenge, what are you doing now, and does that affect different aspects of your life? For example, ask yourself, “How has my life changed at home, at my school, with my friends, or with my family?”

6. What have you done to make your school or your community a better place?

Things to consider: Think of community as a term that can encompass a group, team or a place – like your high school, hometown, or home. You can define community as you see fit, just make sure you talk about your role in that community. Was there a problem that you wanted to fix in your community?

Why were you inspired to act? What did you learn from your effort? How did your actions benefit others, the wider community or both? Did you work alone or with others to initiate change in your community?

7. What is the one thing that you think sets you apart from other candidates applying to the University of California?

Things to consider: Don’t be afraid to brag a little. Even if you don’t think you’re unique, you are — remember, there’s only one of you in the world. From your point of view, what do you feel makes you belong on one of UC’s campuses? When looking at your life, what does a stranger need to understand in order to know you?

What have you not shared with us that will highlight a skill, talent, challenge, or opportunity that you think will help us know you better? We’re not necessarily looking for what makes you unique compared to others, but what makes you, YOU.

MT560 Algebraic Topology Autumn 2016 – Homework 2
1. Suppose that X,Y are spaces with Y Hausdorﬀ and f : X → Y is continuous. Show that the graph of f, Gr(f) = {(x,f(x)) ; x ∈ X}, is a closed subset of X ×Y .
2. Let (X,T) be a topological space and ‘∗’ a ‘point’ not in the set X. Let X0 = X ∪{∗}. (a) Let T0 = T ∪{X0}. Show that T0 is a topology on X0. (b) Show that X is a subspace of X0. (c) Show that (X0,T0) is compact. (d) Suppose X is Hausdorﬀ. Does it follow that X0 is Hausdorﬀ?
3. Alexandroﬀ 1-point compactiﬁcation. Let (X,T) be a topological space and ∗ a ‘point’ not in the set X. Let ˆ X = X ∪{∗}. (a) Let TA := T ∪{U ∪{∗} ; X − U is closed and compact in X}. Show that ( ˆ X,TA) is a compact topological space containing X asa subspace. (Note that X is open in ˆ X.) (b) Suppose that X is Hausdorﬀ and locally compact (i.e. for each x ∈ X, there exists an open set U such that x ∈ U, and its closure, ¯ U, is compact). Deduce that ˆ X is Hausdorﬀ. (c) Suppose X is not compact. Deduce that X is dense in ˆ X. Is this the case if X is compact? (d) Show that the Alexandroﬀ 1-point compactiﬁcation of the open n- cell en is homeomorphic to the n-sphere Sn. Show that attaching the n-disc Dn to a 0-cell ∗, via the constant map f : ∂Dn →{∗}, also gives the n-sphere.

(a) At age 3 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

(b) At age 12 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

(5 pts) Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your final conclusion.

(10 pts)

(a) _______ (fill in the blank)

(b) Let State the exponential form of the equation.

(c) Determine the numerical value of , in simplest form. Work optional.

(10 pts) Let f (x) = 2x2 – x – 10 and g(x) = 3x + 1

(a) Find the composite function and simplify the results. Show work.

(b) Find . Show work.

(15 pts) Let

(a) Find f – 1 , the inverse function of f. Show work.

(b) What is the domain of f ? What is the domain of the inverse function?

(c) What is f (2) ? f (2) = ______ work/explanation optional

(d) What is f – 1 ( ____ ), where the number in the blank is your answer from part (c)? work/explanation optional

(15 pts) Let f (x) = e x – 1 + 4.

Answers can be stated without additional work/explanation.

(a) Which describes how the graph of f can be obtained from the graph of y = ex ? Choice: ________

Shift the graph of y = ex to the left by 1 unit and up by 4 units.

Shift the graph of y = ex to the right by 1 unit and up by 4 units.

Reflect the graph of y = ex across the x-axis and shift up by 4 units.

Reflect the graph of y = ex across the y-axis and shift up by 4 units.

(b) What is the domain of f ?

(c) What is the range of f ?

(d) What is the horizontal asymptote?

(e) What is the y-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth).

(f) Which is the graph of f ?

GRAPH A GRAPH B GRAPH C GRAPH D

NONLINEAR MODELS – For the latter part of the quiz, we will explore some nonlinear models.

(15 pts) QUADRATIC REGRESSION

Data:On a particular summer day, the outdoor temperature was recorded at 8 times of the day, and the following table was compiled. A scatterplot was produced and the parabola of best fit was determined.

REMARKS: The times are the hours since midnight. For instance, 7 means 7 am, and 13 means 1 pm.

(a) Using algebraic techniques we have learned, find the maximum temperaturepredicted by the quadratic model and find the time when it occurred. Report the time to the nearest quarter hour (i.e., __:00 or __:15 or __:30 or __:45). (For instance, a time of 18.25 hours is reported as 6:15 pm.) Report the maximum temperature to the nearest tenth of a degree. Show algebraic work.

(b) Use the quadratic polynomial to estimate the outdoor temperature at 7:30 am, to the nearest tenth of a degree. (work optional)

(c) Use the quadratic polynomial y = -0.3476t2 + 10.948t – 6.0778 together with algebra to estimate the time(s) of day when the outdoor temperature ywas 75 degrees.

(a) Use the exponential function to estimate the temperature difference y when 25 minutes have elapsed. Report your estimated temperature difference to the nearest tenth of a degree. (explanation/work optional)

(b) Since y = C – 69, we have coffee temperature C = y + 69. Take your difference estimate from part (a) and add 69 degrees. Interpret the result by filling in the blank:

When 25 minutes have elapsed, the estimated coffee temperature is ________ degrees.

Some students have a background, identity, interest, talent that is meaningful they believe their application would be incomplete without it. If it sounds like you than please share your story

Amid my school years and as ahead of schedule as in the fourth grade, I was staggered by the assortment of conceivable uses of arithmetic in regular daily existence. The way that even a straightforward thing, for example, leaves developing on a tree, can be put into a succession, a Fibonacci’s arrangement to be more precise, is simply noteworthy. Also, the rundown goes, on with PC frameworks and projects and additionally the web and applications that are all composed and modified utilizing distinctive numerical calculations like for instance Euclid’s calculation. I respect the excellence of head breaking oddities, which sound so natural and self-evident, where as in reality their confirmation is exceptionally troublesome and scientific and utilizes the Horse mystery or Zeno’s conundrum or Grandi’s arrangement. Wherever we turn, whatever we do, there is a math application and that is the manner by which it was since the get-go.

For me by and by arithmetic can clarify a great deal of exceptionally confounded thoughts in an extremely basic manner, and can be utilized to sum them up. Additionally in Mathematics there are a great deal of totally distinctive strategies for accomplishing the same arrangement what makes me much more spurred and inspired by it.

The fulfillment I get from fathoming a scientific errand is comparable to climbing a top of a mountain bluff. The main contrast is that the quantity of these precipices is ceaseless and sky is the breaking point! The themes I have explored the most are, synchronous conditions and Gaussian Elimination. I additionally looked into changed sorts of conditions with whole number arrangements, called Diophantine conditions and in actuality it turned into my investigation point for my IB Higher Level Mathematics Internal Assessment. In my first year of the IB program one of my most loved themes were capacities and separation, and what’s more I have been perusing about Leibniz’s documentations, which clarify a practical relationship between an autonomous and obscene variables and supernatural capacities. A few people like perusing fiction and some like finding out about the history and for me investigating the World of arithmetic is the thing that fulfills me the most.

However difficult the IB program abnormal state arithmetic are , I’ve been taking exams in the Russian secondary school amid each school break throughout the previous four years. I reached a conclusion that the methodology arithmetic is educated in Russia is some way or another not the same as in the school in the UK. This helped me to widen my perspectives and learning significantly all the more, particularly in the zones of unadulterated variable based math and hard trigonometry. I have altogether delighted in the utilization of science in these territories, despite the fact that it was very trying for me to study, in two distinct nations and in two unique dialects throughout the previous four years at the same time.

My enthusiasm for arithmetic is great to the point that last year I joined the Moscow Institute of Physics and Technology separation learning Olympiad school, keeping in mind the end goal to grow my insight into science much more and have the capacity to work and speak with individuals who share a comparable energy for arithmetic as me. This was a chance for me to increase more extensive supposition about various numerical issues furthermore have entry to some school level scientific issues and arrangements. I have totally delighted in this experience and it was an incredible fulfillment for me to acquire information in programming, material science and arithmetic.

I feel satisfied when I am included and adapting more about the extraordinary World of maths. Later on I might want to have the capacity to apply the information I have, all things considered, for instance in displaying PC projects or making new seeking calculations! I am likewise exceptionally inspired by utilizing arithmetic to foresee monetary patterns, and research into computerized reasoning, which could be executed into stock trade and connected financial matters.

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